A remark on covering of compact Kähler manifolds and applications

  • V. V. Hung Tay Bac Univ., Univ. Danang – Univ. Sci. and Education, Vietnam
  • H. N. Quy Tay Bac Univ., Univ. Danang – Univ. Sci. and Education, Vietnam
Keywords: Complex Monge-Amp`ere operator, ω-plurisubharmonic functions, compact K¨ahler manifolds

Abstract

UDC 517.9

Recently, Kolodziej proved that, on a compact Kähler manifold $M,$ the solutions to the complex Monge – Ampére equation with the right-hand side in $L^p,$ $p>1,$ are Hölder continuous with the exponent depending on $M$ and $\|f\|_p$ (see [Math. Ann., 342, 379-386 (2008)]).
Then, by the regularization techniques in
[J. Algebraic Geom., 1, 361-409 (1992)], the authors in [J. Eur. Math. Soc., 16, 619-647 (2014)] have found the optimal exponent of the solutions.
In this paper, we construct a cover of the compact Kähler manifold $M$ which only depends on curvature of $M.$ Then, as an application, base on the arguments in
[Math. Ann., 342, 379-386 (2008)], we show that the solutions are Hölder continuous with the exponent just depending on the function $f$ in the right-hand side and upper bound of curvature of $M.$

 

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Published
22.01.2021
How to Cite
HungV. V., and QuyH. N. “A Remark on Covering of Compact Kähler Manifolds and Applications”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 73, no. 1, Jan. 2021, pp. 138 -48, doi:10.37863/umzh.v73i1.6038.
Section
Research articles